2cos (problem 3.3.5)

Average Error: 39.7 → 0.4

Time: 21.5s

Precision: binary64

Cost: 39360

Math

\[\cos \left(x + \varepsilon\right) - \cos x \]

↓

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \left(t_0 \cdot \mathsf{fma}\left(\sin x, \cos \left(0.5 \cdot \varepsilon\right), t_0 \cdot \cos x\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))))
   (* (* t_0 (fma (sin x) (cos (* 0.5 eps)) (* t_0 (cos x)))) -2.0)))

C

double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}

↓

double code(double x, double eps) {
	double t_0 = sin((0.5 * eps));
	return (t_0 * fma(sin(x), cos((0.5 * eps)), (t_0 * cos(x)))) * -2.0;
}

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

↓

function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	return Float64(Float64(t_0 * fma(sin(x), cos(Float64(0.5 * eps)), Float64(t_0 * cos(x)))) * -2.0)
end

Wolfram

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] * N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Initial program 39.7

   \[\cos \left(x + \varepsilon\right) - \cos x \]

2. Applied egg-rr 14.8

   \[\leadsto \color{blue}{\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \cdot -2} \]

3. Taylor expanded in x around -inf 14.8

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot -2 \]

4. Applied egg-rr 0.4

   \[\leadsto \left(\color{blue}{\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot 2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(x \cdot 2\right) \cdot 0.5\right)\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot -2 \]

5. Applied egg-rr 0.4

   \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sin x, \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos x\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot -2 \]

6. Final simplification 0.4

   \[\leadsto \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\sin x, \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos x\right)\right) \cdot -2 \]

Alternatives

Alternative 1
Error	0.4
Cost	33088

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ -2 \cdot \left(t_0 \cdot \left(t_0 \cdot \cos x + \sin x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right)\right) \end{array} \]

Alternative 2
Error	0.7
Cost	32840

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ t_1 := \cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -1.6957635392352122:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 62461.521544933064:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(\sin x + t_0 \cdot \cos \left(0.5 \cdot \left(x \cdot 2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.7
Cost	32840

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ t_1 := \cos x \cdot \cos \varepsilon\\ t_2 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.6957635392352122:\\ \;\;\;\;t_1 - \left(\cos x + t_2\right)\\ \mathbf{elif}\;\varepsilon \leq 62461.521544933064:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(\sin x + t_0 \cdot \cos \left(0.5 \cdot \left(x \cdot 2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - t_2\right) - \cos x\\ \end{array} \]

Alternative 4
Error	14.0
Cost	26816

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ -2 \cdot \left(t_0 \cdot \left(t_0 + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right)\right) \end{array} \]

Alternative 5
Error	14.1
Cost	26688

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ -2 \cdot \left(t_0 \cdot \left(\sin x + t_0 \cdot \cos \left(0.5 \cdot \left(x \cdot 2\right)\right)\right)\right) \end{array} \]

Alternative 6
Error	14.8
Cost	13888

\[-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)\right) \]

Alternative 7
Error	14.6
Cost	13640

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -1.6957635392352122:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 8.548859651221712 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	14.8
Cost	13632

\[-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right)\right) \]

Alternative 9
Error	14.7
Cost	13256

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -1.6957635392352122:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 8.548859651221712 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	15.2
Cost	7496

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -1.6957635392352122:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 8.548859651221712 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	20.9
Cost	7372

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -1.6957635392352122:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.853947697536695 \cdot 10^{-46}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0015213690037734044:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.041666666666666664, -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	21.0
Cost	6988

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -1.6957635392352122:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.853947697536695 \cdot 10^{-46}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 8.548859651221712 \cdot 10^{-7}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	34.1
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -1.6957635392352122:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 8.548859651221712 \cdot 10^{-7}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	50.3
Cost	320

\[-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) \]

Reproduce

herbie shell --seed 2022318 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))